On Lie Ideals and Generalized (θΦ Φ)-Derivations in Prime Rings

Abstract

Let R be a prime ring with characteristic different from two and S a non-empty subset of R. Suppose that θ, Φ are endomorphisms of R. An additive mapping F : R → R is called a generalized (θ, Φ)-derivation (resp. generalized Jordan (θ, Φ)-derivation) on S if there exists a (θ, Φ)-derivation d : R → R such that F(xy) = F(x)θ(y) + Φ(x)d(y) (resp. F(x ²) = F(x)θ(x) + Φ(x)d(x)), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u ² ∈ U, for all u ∈ U. In the present paper, it is shown if θ is an automorphism of R then every generalized Jordan (θ, Φ)-derivation F on U is a generalized (θ, Φ)-derivation on U.

Key Words:

• Prime rings
• Lie ideals
• Torsion free rings
• Derivations
• Generalized derivations
• Generalized (θ, Φ)-derivations
• Generalized Jordan (θ, Φ)-derivations

2000 AMS Subject Classification:

• 16W25
• 16N60
• 16U80

Acknowledgments

The authors are greatly indebted to the referee for his/her several useful suggestions and valuable comments. Also, the third author gratefully acknowledges the financial support he received from U.G.C. India for this research.

Notes

^#Communicated by N. Gupta.